/*
 * zzllrr Mather
 * zzllrr@gmail
 * Released under MIT License
 */

wiki['Formula/Sequence/Sum']=


detail('等差或等比数列求和公式',Table([ZLR('前n项求和 记法 结果')],[
	[['等差数列','a+(a+d)+(a+2d)','+⋯+(a+(n−1)d)'].join(kbr),
		Eq([sum('i',1,'n','(a+(i-1)d)','',''), 'na+d'+sum('i',1,'n-1','i','','')]),
		Eq([frac('(a + a_n)n',2,''),'na + '+frac('n(n-1)',2,'')+'d'])+kbr+kxf('Gauss')+': 数列逆序，分别相加'
	],
	[zlr('','1 2 3 ⋯ n','+'),sum('i',1,'n','i','',''),frac(1,2,'')+'n(n+1)'],

	['等比数列 q≠1'+kbr+ZLR3('a',ZLR(' q q ⋯ q'),ZLR('  ^2  ^{n-1}'),'+'),
		Eq([sum('i',1,'n','aq^{i-1}','',''), 'a'+sum('i',0,'n-1','q^i','','')]),
		Eq([frac('a - a_nq','1-q',''),frac('a(1-q^n)','1-q','')])+kbr+'数列×q，错位相减'
	],
	['q≠1'+kbr+'1+q+q^2+⋯+q^{n-1}',
		Eq([sum('i',1,'n','q^{i-1}','',''), sum('i',0,'n-1','q^i','','')]),
		Eq([frac('1 - a_{n+1}','1-q',''),frac('1-q^n','1-q','')])+kbr+'数列×q，错位相减'
	],

	[zlr('','2 4 6 ⋯ 2n','+'),
	sum('i',1,'n','2i','','')+'\\\\=2'+sum('i',1,'n','i','',''),
	'n(n+1)'
	],

	[zlr('','1 3 5 \\\\⋯ (2n-1)','+'),
		[sum('i',1,'n','(2i-1)','',''),
		sum('i',1,'2n','i','','')+'-'+sum('i',1,'n','2i','','')+'(方法1)',
		2+sum('i',1,'n','i','','')+'-'+sum('i',1,'n','1','','')+'(方法2)',
		].join(kbr+'='),
		['n^2','方法1: 2n项之和减去n项和','方法2: 拆开通项多项式（下同）'].join(kbr)
	],

],'wiki').replace(/\n/g,br))+

detail('等幂和公式',Table([ZLR('前n项求和 记法 结果')],[

	[zlr('','1 2 3 ⋯ n','+'),
		Eq([sum('i',1,'n','i','',''),'a（记作）']),
		frac(1,2,'')+'n(n+1)'
	],
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^2','+'),
		Eq([sum('i',1,'n','i^2','',''),
			frac('n+1/2',3,'')+'2a',
			frac('2n+1',3,'')+'a',
		]),
		Eq([frac(1,6,'')+'n(n+1)(2n+1)',
			frac(1,3,'')+'n(n+1)'+zp('n+'+frac(1,2,''))
		])+kbr+
		'可利用(n+1)^3-n^3=3n^2+3n+1 累加相消'+kbr+
		'也可利用排列组合公式i^2=i(i-1)+i=2C_i^2+C_i^1'+kbr+
		'i=1,2,⋯,n，累加得到'+kbr+
		'=2C_{n+1}^3+C_{n+1}^2'
	],
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^3','+'),
		Eq([sum('i',1,'n','i^3','',''),
			'a^2'
		]),
		Eq([frac(1,4,'')+'n^2(n+1)^2',
			'(1+2+3+⋯+n)^2',
			
		])+kbr+
		'\\text{'+gM("Nicomachus's theorem")+'，也可利用面积可视化证明}'+kbr+
		'也可利用排列组合公式'+kbr+
		Eq(['i^3=i(i-1)(i-2)+3i(i-1)+i',
			'3!C_i^3+3⋅2!C_i^2+C_i^1',
		])+kbr+'累加得到'+kbr+'3!C_{n+1}^4+3⋅2!C_{n+1}^3+C_{n+1}^2'

	],
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^4','+'),
		Eq([sum('i',1,'n','i^4','',''),
			frac('n+1/2',5,'')+frac('1',3,'')+'(12a^2-2a)',
			frac('2n+1',15,'')+'a(6a-1)',
		]),
		Eq([frac(1,30,'')+'n(n+1)(2n+1)(3n^2+3n-1)',
			frac(1,5,'')+'n(n+1)'+zp('n+'+frac(1,2,''))+zp('n^2+n-'+frac(1,3,''))

		])+kbr+
		'也可利用排列组合公式'+kbr+Eq(['i^4=i(i-1)(i-2)(i-3)+6i(i-1)(i-2)+7i(i-1)+i',
		'4!C_i^4+6⋅3!C_i^3+7⋅2!C_i^2+C_i^1',
		])+kbr+'累加得到，'+
		'4!C_{n+1}^5+6⋅3!C_{n+1}^4+7⋅2!C_{n+1}^3+C_{n+1}^2'
	],
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^5','+'),
		Eq([sum('i',1,'n','i^5','',''),
			frac('a^2',3,'')+'(4a-1)',
		]),
		Eq([frac(1,12,'')+'n^2(n+1)^2(2n^2+2n-1)',
			frac(1,6,'')+'n^2(n+1)^2'+zp('n^2+n-'+frac(1,2,'')),
			
		])+kbr+
		'也可利用排列组合公式i^5=i(i-1)(i-2)(i-3)(i-4)+'+kbr+'(1+2+3+4)i(i-1)(i-2)(i-3)+'+kbr+'\\frac{3^4-(2^5-2)-1}{2!}i(i-1)(i-2)+'
		+kbr+'(2^4-1)i(i-1)+i'+kbr+
		'=5!C_i^5+C_5^2⋅4!C_i^4+25⋅3!C_i^3+15⋅2!C_i^2+C_i^1'+kbr+
		'累加得到，5!C_{n+1}^6+C_5^2⋅4!C_{n+1}^5+25⋅3!C_{n+1}^4+15⋅2!C_{n+1}^3+C_{n+1}^2'  
	],
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^6','+'),
		Eq([sum('i',1,'n','i^6','',''),
			frac('n+1/2',7,'')+frac('1',3,'')+'(24a^3-12a^2+2a)',
			frac('2n+1',21,'')+'a(12a^2-6a+1)',
			frac('2n+1',21,'')+'a(12a^2-(6a-1))',
		]),
		Eq([frac(1,42,'')+'n(n+1)(2n+1)(3n^4+6n^3-3n+1)',
			frac(1,7,'')+'n(n+1)'+zp('n+'+frac(1,2,''))+zp('n^4+2n^3-n+'+frac(1,3,'')),
			
		])+kbr+
		'也可利用排列组合公式'+kbr+'i^6='+
		'6!C_i^6+C_6^2⋅5!C_i^5+\\frac{4^5-90⋅3!-31⋅2!⋅3/2-1}{3!}⋅4!C_i^4+'+kbr+'\\frac{3^5-31⋅2!-1}{2}⋅3!C_i^3+(2^5-1)⋅2!C_i^2+C_i^1'+kbr+
		'累加得到，'+kbr+
		'6!C_{n+1}^7+C_6^2⋅5!C_{n+1}^6+65⋅4!C_{n+1}^5+90⋅3!C_{n+1}^4+31⋅2!C_{n+1}^3+C_{n+1}^2'
		
	],
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^7','+'),
		Eq([sum('i',1,'n','i^7','',''),
			frac('a^2',3,'')+'(6a^2-4a+1)',
		]),
		Eq([frac(1,24,'')+'n^2(n+1)^2(3n^4+6n^3-n^2-4n+2)',
			frac(1,8,'')+'n^2(n+1)^2'+ zp('n^4+2n^3-'+frac(1,3,'')+'n^2-'+frac(4,3,'')+'n+'+frac(2,3,'')),
			
		])
	],
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^8','+'),
		Eq([sum('i',1,'n','i^8','',''),
			frac('n+1/2',9,'')+frac('1',5,'')+'(80a^4-80a^3+36a^2-6a)',
			frac('2n+1',45,'')+'a(40a^3-40a^2+18a-3)',
			frac('2n+1',45,'')+'a(40a^2(a-1)+3(6a-1))',
		]),
		Eq([frac(1,90,'')+'n(n+1)(2n+1)(5n^6+15n^5+5n^4-15n^3-n^2+9n-3)',
			frac(1,9,'')+'n(n+1)'+zp('n+'+frac(1,2,''))+ zp('n^6+3n^5+n^4-3n^3-'+frac(1,5,'')+'n^2+'+frac(9,5,'')+'n-'+frac(3,5,''))
		])
	],
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^9','+'),
		Eq([sum('i',1,'n','i^9','',''),
			frac('a^2',5,'')+'(16a^3-20a^2+12a-3)',
		]),
		Eq([frac(1,20,'')+'n^2(n+1)^2(n^2+n-1)(2n^4+4n^3-n^2-3n+3)',
			frac(1,10,'')+'n^2(n+1)^2(n^2+n-1)'+ zp('n^4+2n^3-'+frac(1,2,'')+'n^2-'+frac(3,2,'')+'n+'+frac(3,2,'')),
			
		])
	],
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^{10}','+'),
		Eq([sum('i',1,'n','i^{10}','',''),

			frac('n+1/2',11,'')+frac('1',3,'')+'(96a^5-160a^4+136a^3-60a^2+10a)',
			frac('2n+1',33,'')+'a(48a^4-80a^3+68a^2-30a+5)',
			frac('2n+1',33,'')+'a(4a^2(12a^2-20a+17)-5(6a-1))',
		]),
		Eq([frac(1,66,'')+'n(n+1)(2n+1)(n^2+n-1)(3n^6+9n^5+2n^4-11n^3+3n^2+10n-5)',
			frac(1,11,'')+'n(n+1)'+zp('n+'+frac(1,2,''))+'(n^2+n-1)'+ zp('n^6+3n^5+'+frac(2,3,'')+'n^4-'+frac(11,3,'')+'n^3+n^2+'+frac(10,3,'')+'n-'+frac(5,3,''))
		])
	],
	
	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^{11}','+'),
		Eq([sum('i',1,'n','i^{11}','',''),
			
			frac('a^2',3,'')+'(16a^4-32a^3+34a^2-20a+5)',
		]),
		Eq([frac(1,24,'')+'n^2(n+1)^2(2 n^8 + 8 n^7 + 4 n^6 - 16 n^5 - 5 n^4 + 26 n^3 - 3 n^2 - 20 n + 10)',
			frac(1,12,'')+'n^2(n+1)^2'+zp('n^8 + 4 n^7 + 2 n^6 - 8 n^5 - 5\\/2 n^4 + 13 n^3 - 3\\/2 n^2 - 10 n + 5'),		
		])
	],

],'wiki').replace(/\n/g,br))+

detail('等幂和通用公式',Table([ZLR('前n项求和 记法 结果')],[

	[ZLR3('',ZLR('1 2 3 ⋯ n'),'^m','+'),
		Eq(['f(m,n) ~其中m是等幂次，n项和', sum('i',1,'n','i^m','',''),
			piece([
				['n','m=0'],
				['a','奇数m=1'],
				[frac(4,'m+1','')+'a^2⋯','奇数m>3'],
				[frac(2,'m+1','')+'a(2n+1)⋯','偶数m>0']
			])
		])+kbr+
		'另外，利用'+kbr+Eq(['(n+1)^{k+1}-1',
			sum('m',1,'n','((m+1)^{k+1}-m^{k+1})','',''),
			sum('m',1,'n',sum('i',0,'k',binom('k+1','i')+'m^i','',''),'',''),
			sum('i',0,'k',binom('k+1','i')+sum('m',1,'n','m^i','',''),'',''),
			sum('i',0,'k',binom('k+1','i')+zp(ZLR3('',ZLR('1 2 3 ⋯ n'),'^i','+')),'',''),
		])+kbr+
		'则得到递推式①'+kbr+'f(k+1,n+1)-1='+sum('i',0,'k+1',binom('k+1','i')+'f(i,n)','','')+kbr+
		'另外，利用(n+1)^k-n^k='+sum('i',0,'k-1',binom('k','i')+'n^i','','')+kbr+'，n=1,2,⋯累加相消'+kbr+
		'则得到关系式②'+kbr+'(n+1)^k-1='+sum('i',0,'k-1',binom('k','i')+'f(i,n)','','')+'='+zp(sum('i',0,'k',binom('k','i')+'f(i,n)','',''))+'-f(k,n)'+kbr+
		'则③ ~ f(k,n+1)-1='+sum('i',0,'k',binom('k','i')+'f(i,n)','','')+kbr+
		'（事实上，也可令①式中的k+1变成k，立即可得）'+kbr+
		'①-③，得'+kbr+
		'f(k+1,n+1)-f(k,n+1)='+sum('i',1,'k+1',binom('k','i-1')+'f(i,n)','','')+'='+sum('i',0,'k',binom('k','i')+'f(i+1,n)','',''),

		Eq([piece([
				['{π^2}\\/6','m=-2，n→∞'],
				['\\ln n+γ = \\ln (e^γn)','m=-1，n→∞'],
				['n','m=0'],
				[frac(1,'2','')+'n(n+1)','奇数m=1'],
				[frac(1,'m+1','')+'n^2(n+1)^2⋯','奇数m>3'],
				[frac(1,'m+1','')+'n(n+1)(2n+1)⋯','偶数m>0']
			]),
			'1\\/{m+1}'+sum('j',0,'m',binom('m+1','j')+'(-1)^jB^-_jn^{m+1-j}','',''),
			'1\\/{m+1}'+sum('j',0,'m',binom('m+1','j')+'B^+_jn^{m+1-j}','',''),
			'1\\/{m+1}'+sum('j',0,'m',binom('m+1','j')+'B^jn^{m+1-j}','',''),
			frac('(\\bold B+n)^{m+1}-\\bold B^{m+1}','m+1',''),

			'm!'+sum('j',0,'m','\\frac{B^+_jn^{m+1-j}}{j!(m+1-j)!}','',''),
			frac('n^{m+1}','m+1','')+'+'+frac('n^{m}','2','')+'+'+'1\\/{m+1}'+sum('j',2,'m',binom('m+1','j')+'B_jn^{m+1-j}','',''),


			
		])+kbr+kxf(' 阴影形式 Umbral form')+'将B下标替换为上标'+kbr+
		' (1-2\\bold {B} )^{m}=(2-2^{m})B_{m}'+kbr+

		'也可利用排列组合公式'+kbr+'i^m='+
		'm!C_i^m+C_m^2⋅(m-1)!C_i^{m-1}+⋯+'+kbr+
		'(4^m-C_4^33^m+C_4^22^m-C_4^1)⋅C_i^4+(3^m-C_3^22^m+C_3^1)⋅C_i^3+(2^m-C_2^1)⋅C_i^2+C_i^1'+kbr+
		'=A_i^m+C_m^2⋅A_i^{m-1}+⋯+'+kbr+
		'(4^m-C_4^33^m+C_4^22^m-C_4^1)A_i^4/4!+(3^m-C_3^22^m+C_3^1)A_i^3/3!+(2^m-C_2^1)A_i^2/2!+A_i^1'+kbr+

		'='+sum('k',1,'m',zp('k^m-C_k^1(k-1)^m+C_k^2(k-2)^m+⋯+(-1)^{k+1}C_k^1')+'C_i^k','','')+kbr+
		'='+sum('k',1,'m',zp(sum('j',1,'k','(-1)^{k+j}C_k^jj^m','',''))+'C_i^k','','')+kbr+

		'观察上式，令其中k=m时，易得m!='+sum('j',1,'m','(-1)^{m+j}C_m^jj^m','','')+kbr+
		'则(m-1)!='+sum('j',1,'m-1','(-1)^{m+j-1}C_{m-1}^jj^{m-1}','','')+' ~①'+kbr+
		'而令其中k=m-1时，易得C_m^2⋅(m-1)!='+sum('j',1,'m-1','(-1)^{m+j-1}C_{m-1}^jj^m','','')+' ~②'+kbr+
		'可发现①左边乘以C_m^2，相当于右边乘以j（多项式中都升一次幂）'+kbr+
		'另外②-①，得到'+kbr+
		'(m-2)(m+1)(m-1)!/2='+sum('j',2,'m-1','(-1)^{m+j-1}C_{m-1}^j(j-1)j^{m-1}','','')+kbr+
		
		'i=1,2,⋯,n累加得到，'+kbr+
		'f(m,n)=m!C_{n+1}^{m+1}+C_m^2⋅(m-1)!C_{n+1}^m+⋯+'+kbr+
		'(5^m-C_5^44^m+C_5^33^m-C_5^22^m+C_5^1)⋅C_{n+1}^6+'+kbr+
		'(4^m-C_4^33^m+C_4^22^m-C_4^1)⋅C_{n+1}^5+(3^m-C_3^22^m+C_3^1)⋅C_{n+1}^4+(2^m-C_2^1)⋅C_{n+1}^3+C_{n+1}^2'+kbr+
		'='+sum('k',1,'m',zp('k^m-C_{k}^{k-1}(k-1)^m+C_{k}^{k-2}(k-2)^m+⋯+(-1)^{k-1}C_{k}^1')+'C_{n+1}^{k+1}','','')+kbr+
		'='+sum('k',1,'m',zp('k^m-C_{k}^1(k-1)^m+C_{k}^2(k-2)^m+⋯+(-1)^{k-1}C_{k}^1')+'C_{n+1}^{k+1}','','')+kbr+
		'='+sum('k',1,'m',zp(sum('j',1,'k','(-1)^{k+j}C_{k}^jj^m','',''))+'C_{n+1}^{k+1}','','')+kbr


			
	],



	['m奇数时'+kbr+ZLR3('',ZLR('1 2 3 ⋯ n'),'^m','+'),
		sum('i',1,'n','i^m','',''),
		Eq([
			'c_1a^2+c_2a^3+ ⋯ + c_{\\tfrac{m-1}2}a^{\\tfrac{m+1}2}',
			'a'+zp('c_1a+c_2a^2+ ⋯ + c_{\\tfrac{m-1}2}a^{\\tfrac{m-1}2}'),
			frac('1','2^{m+1}(m+1)','')+sum('j',0,'\\tfrac{m-1}{2}',binom('m+1','2j')+'(2-2^{2j})B_{2j}\\[(8a+1)^{\\frac{m+1}{2}-j}-1\\]','',''),
			frac('1','2^{m}(m+1)','')+sum('j',0,'\\tfrac{m-1}{2}',binom('m+1','2j')+'(1-2^{2j-1})B_{2j}\\[(8a+1)^{\\frac{m+1}{2}-j}-1\\]','',''),
		]),

	],


	['m偶数时'+kbr+ZLR3('',ZLR('1 2 3 ⋯ n'),'^m','+'),
		sum('i',1,'n','i^m','',''),

		Eq([
			frac('n+1/2','m+1','')+zp('c_12a+c_23a^2+ ⋯ + c_{\\tfrac{m}2}(\\tfrac{m}2+1)a^{\\tfrac{m}2}'),
			frac('n+1/2','m+1','')+'a'+zp('2c_1+3c_2a+ ⋯ + (\\tfrac{m}2+1)c_{\\tfrac{m}2}a^{\\tfrac{m}2-1}'),
			frac('1','2^{m-1}(m+2)','')+sum('j',0,'\\tfrac{m}{2}',binom('m+2','2j')+'(m+2-2j)(1-2^{2j-1})B_{2j}(8a+1)^{\\frac{m}{2}-j}','',''),
			
		])+kbr+'注意：第一个式子后面括号，是对m+1时公式中的a求导'+kbr+"f(m)="+frac('n+1/2','m+1','')+"f'(m+1)",

	],

	['等奇次幂和'+kbr+ZLR3('',ZLR('1 2 3 ⋯ n'),'^{2m+1}','+'),
		sum('i',1,'n','i^{2m+1}','',''),
		Eq([
			'c_1a^2+c_2a^3+ ⋯ + c_{m}a^{m+1}',
			'a'+zp('c_1a+c_2a^2+ ⋯ + c_{m}a^{m}'),
			frac('1','2^{2m+2}(2m+2)','')+sum('j',0,'m',binom('2m+2','2j')+'(2-2^{2j})B_{2j}\\[(8a+1)^{m+1-j}-1\\]','',''),
			frac('1','2^{2m+1}(2m+2)','')+sum('j',0,'m',binom('2m+1','2j')+'(1-2^{2j-1})B_{2j}\\[(8a+1)^{m+1-j}-1\\]','',''),
			frac('1','2m+2','')+sum('j',0,'2m+1',binom('2m+2','j')+'B_{j}n^{2m+2-j}','',''),
		]),

	],


	['等偶次幂和'+kbr+ZLR3('',ZLR('1 2 3 ⋯ n'),'^{2m}','+'),
		sum('i',1,'n','i^{2m}','',''),

		Eq([
			frac('n+1/2','2m+1','')+zp('c_12a+c_23a^2+ ⋯ + c_{m}(m+1)a^{m}'),
			frac('n+1/2','2m+1','')+'a'+zp('2c_1+3c_2a+ ⋯ + (m+1)c_{m}a^{m-1}'),
			frac('1','2^{2m}(m+1)','')+sum('j',0,'m',binom('2m+2','2j')+'(m+1-j)(2-2^{2j})B_{2j}(8a+1)^{m-j}','',''),
			frac('1','2^{2m-1}(m+1)','')+sum('j',0,'m',binom('2m+1','2j')+'(m+1-j)(1-2^{2j-1})B_{2j}(8a+1)^{m-j}','',''),
			frac('1','2m+1','')+sum('j',0,'2m',binom('2m+1','j')+'B_{j}n^{2m+1-j}','',''),
			
		])+kbr+'注意：第一个式子后面括号，是对2m+1时公式中的a求导'+kbr+"f(m)="+frac('n+1/2','2m+1','')+"f'(m+1)",

	],


],'wiki').replace(/\n/g,br))+

detail('奇偶项等幂和公式',Table([ZLR('前n项求和 记法 结果')],[

	[zlr('','2 4 6 ⋯ 2n','+'),
	sum('i',1,'n','2i','','')+'\\\\=2'+sum('i',1,'n','i','',''),
	'n(n+1)'
	],

	[zlr('','1 3 5 \\\\⋯ (2n-1)','+'),
		[sum('i',1,'n','(2i-1)','',''),
		sum('i',1,'2n','i','','')+'-'+sum('i',1,'n','2i','','')+'(方法1)',
		2+sum('i',1,'n','i','','')+'-'+sum('i',1,'n','1','','')+'(方法2)',
		].join(kbr+'='),
		['n^2','=n(2n+1)-n(n+1)','=n(n+1)-n','方法1: 2n项之和减去n项和','方法2: 拆开通项多项式（下同）'].join(kbr)
	],

	[ZLR3('',ZLR('1 3 5 ⋯ (2n-1)'),'^2','+').replace('⋯',kbr+'⋯'),
		Eq([sum('i',1,'n','(2i-1)^2','',''),
			sum('i',1,'2n','i^2','','')+'-'+sum('i',1,'n','(2i)^2','','')
		]),
		
		frac(1,3,'')+'n(2n-1)(2n+1)'+kbr+'='+frac(1,3,'')+'n(4n^2-1)'
	],
	[ZLR3('',ZLR('1 3 5 ⋯ (2n-1)'),'^3','+').replace('⋯',kbr+'⋯'),
		Eq([sum('i',1,'n','(2i-1)^3','',''),
			sum('i',1,'2n','i^3','','')+'-'+sum('i',1,'n','(2i)^3','','')
		]),
		
			frac(1,4,'')+'(2n)^2(2n+1)^2-'+frac(1,4,'')+'2^3n^2(n+1)^2'+kbr+
			'=n^2(2n^2-1)'
		
	],
	[ZLR3('',ZLR('1 3 5 ⋯ (2n-1)'),'^4','+').replace('⋯',kbr+'⋯'),
		Eq([sum('i',1,'n','(2i-1)^4','',''),
			sum('i',1,'2n','i^4','','')+'-2^4'+sum('i',1,'n','i^4','','')
		]),
		frac(1,30,'')+'2n(2n+1)(4n+1)(12n^2+6n-1)'+kbr+
		'-'+frac('2^4',30,'')+'n(n+1)(2n+1)(3n^2+3n-1)'+kbr+

		
		'='+frac(1,15,'')+'n(2n+1)(24n^3-12n^2-14n+7)'
	],
	[ZLR3('',ZLR('1 3 5 ⋯ (2n-1)'),'^5','+').replace('⋯',kbr+'⋯'),
		Eq([sum('i',1,'n','(2i-1)^5','',''),
			sum('i',1,'2n','i^5','','')+'-2^5'+sum('i',1,'n','i^5','','')
		
		]),//bigintsim('2^2n^2((4n^2+4n+1)(8n^2+4n-1)-(n^2+2n+1)(16n^2+16n-8))','',{'n':2})/12n/7n
		frac(1,3,'')+'n^2(2n^2-1)()'
	],
	['奇数项等幂和'+kbr+ZLR3('',ZLR('1 3 5 ⋯ (2n-1)'),'^m','+').replace('⋯',kbr+'⋯'),
		Eq([sum('i',1,'n','(2i-1)^m','',''),
			sum('i',1,'2n','i^m','','')+'-2^m'+sum('i',1,'n','i^m','',''),
			'f_m(2n)-2^mf_m(n)',
		]),
		piece([
			[frac(1,'2','')+'n(n+1)','奇数m=1'],
			[frac(1,'m+1','')+'n^2(n+1)^2⋯','奇数m>3'],
			[frac(1,'m+1','')+'n(n+1)(2n+1)⋯','偶数m']
		])
	],




],'wiki').replace(/\n/g,br))+

detail('交错等幂和公式',Table([ZLR('前n项求和 记法 结果')],[

	[['1-2+3-⋯','+(-1)^{n-1}n'].join(kbr),
		sum('i',1,'n','(-1)^{i-1}⋅i','',''),
		Eq([piece([[frac(1,2,'')+'(n+1)','n为奇数'],['-'+frac('n',2,''),'n为偶数']]),
			frac('1+(-1)^{n-1}(2n+1)',4,''),
		])+
		
		kbr+Eq(['利用F(n)='+piece(['f(n) & n为奇数','g(n) & n为偶数']),
			frac('f+g',2,'')+'-(\\cos nπ)'+frac('f-g',2,''),
			frac('f+g',2,'')+'-(-1)^n'+frac('f-g',2,''),
			frac('f+g',2,'')+'+(-1)^{n-1}'+frac('f-g',2,''),
		])
	],


	[['1^2-2^2+3^2-⋯','+(-1)^{n-1}n^2'].join(kbr),
		sum('i',1,'n','(-1)^{i-1}⋅i^2','',''),
		
		'(-1)^{n-1}'+frac(1,2,'')+'n(n+1)'
	],

	[['1^3-2^3+3^3-⋯','+(-1)^{n-1}n^3'].join(kbr),
		sum('i',1,'n','(-1)^{i-1}⋅i^3','',''),
		
		frac('-1+(-1)^{n-1}(4n^3+6n^2-1)',8,''),
	],

	[['1^4-2^4+3^4-⋯','+(-1)^{n-1}n^4'].join(kbr),
		sum('i',1,'n','(-1)^{i-1}⋅i^4','',''),
		
		'(-1)^{n-1}'+frac(1,2,'')+'n(n+1)(n^2+n-1)',
		
	],

	[['1^m-2^m+3^m-⋯','+(-1)^{n-1}n^m'].join(kbr),
		sum('i',1,'n','(-1)^{i-1}⋅i^m','',''),
		[
		'(-1)^{n-1}'+frac(1,2,'')+'n(n+1)(????)+',
			'\\left(0(偶)或(-1)^{(m-1)/2}'+frac(1,'2^{m-1}','')+'(奇)\\right)',
		].join(kbr)
	],

],'wiki').replace(/\n/g,br))+


detail('连续数乘积和公式',Table([ZLR('前n项求和 记法 结果')],[

	['1⋅2+2⋅3+3⋅4\\\\+⋯+n(n+1)',
		Eq([sum('i',1,'n','i(i+1)','',''),
			'①多项式方法'+sum('i',1,'n','i^2','','')+'+'+sum('i',1,'n','i','',''),
			'②裂项'+frac(1,3,'')+sum('i',1,'n',zp('i(i+1)(i+2)-(i-1)i(i+1)'),'','')
		]),
		frac(1,3,'')+'n(n+1)(n+2)'
	],
	['1⋅2⋅3+2⋅3⋅4+3⋅4⋅5\\\\+⋯+n(n+1)(n+2)',
		sum('i',1,'n','i(i+1)(i+2)','',''),
		frac(1,4,'')+'n(n+1)(n+2)(n+3)'
	],
	['1⋅2⋅3⋅4+2⋅3⋅4⋅5\\\\+⋯+n(n+1)(n+2)(n+3)',
		sum('i',1,'n','i(i+1)(i+2)(i+3)','',''),
		frac(1,5,'')+'n(n+1)(n+2)(n+3)(n+4)'
	],
	['1⋅2⋅3⋅4⋅5+2⋅3⋅4⋅5⋅6\\\\+⋯+n(n+1)(n+2)(n+3)(n+4)',
		sum('i',1,'n','i(i+1)(i+2)(i+3)(i+4)','',''),
		frac(1,6,'')+'n(n+1)(n+2)(n+3)(n+4)(n+5)'
	],
	[['1⋅2⋅3⋯k+','2⋅3⋅4⋯(k+1)+','⋯+','n(n+1)(n+2)⋯(n+k-1)'].join(kbr),
		Eq([sum('i',1,'n','i(i+1)⋯(i+k-1)','',''),
			'①'+sum('i',1,'n',prod('j',1,'k','(i+j-1)','',''),'',''),
			'②'+frac(1,'k+1','')+sum('i',1,'n','i(i+1)⋯(i+k)-(i-1)i⋯(i+k-1)','',''),
		]),
		
		[frac(1,'k+1','')+'n(n+1)(n+2)','(n+3)⋯(n+k)',
			'='+frac(1,'k+1','')+prod('i',1,'k+1','(n+i-1)','',''),
			'='+frac(1,'k+1','')+frac('(n+k)!','(n-1)!',''),
			'='+frac(1,'k+1','')+'A^{k+1}_{n+k}',
		].join(kbr)
	],


],'wiki').replace(/\n/g,br))+

detail('等差⋅等比 内积求和',Table([ZLR('前n项求和 记法 结果')],[


	[['连续数⋅等比','q+2⋅q^2+3⋅q^3+⋯+n⋅q^n'].join(kbr),
		sum('i',1,'n','i⋅q^i','','')+kbr+
		'(1-q)'+sum('i',1,'n','i⋅q^i','','')+'='+zp(sum('i',1,'n','q^i','',''))+'-n⋅q^{n+1}',
		[
		Eq([frac(zp(sum('i',1,'n','q^i','',''))+'-n⋅q^{n+1}','1-q',''),
		frac('q','1-q','')+'⋅'+zp(frac('1-q^n','1-q','')+'-nq^n'),
		]),
		'数列×q，错位相减'
		].join(kbr)
	],

	[['','1+2⋅q+3⋅q^2+⋯+n⋅q^{n-1}'].join(kbr),
		sum('i',1,'n','i⋅q^{i-1}','','')+kbr+
		'(1-q)'+sum('i',1,'n','i⋅q^{i-1}','','')+'='+zp(sum('i',1,'n','q^{i-1}','',''))+'-n⋅q^{n}',
		[
		Eq([frac(zp(sum('i',1,'n','q^{i-1}','',''))+'-n⋅q^{n}','1-q',''),
		frac('1','1-q','')+'⋅'+zp(frac('1-q^n','1-q','')+'-nq^n'),
		]),
		'数列×q，错位相减',
		].join(kbr)
	],
	[['','1+2⋅2+3⋅2^2+⋯+n⋅2^{n-1}'].join(kbr),
		sum('i',1,'n','i⋅2^{i-1}','','')+kbr+
		'(1-2)'+sum('i',1,'n','i⋅2^{i-1}','','')+'='+zp(sum('i',1,'n','2^{i-1}','',''))+'-n⋅2^{n}',
		[
		Eq(['n⋅2^{n}-'+sum('i',1,'n','2^{i-1}','',''),
		'(n-1)2^n+1',
		]),
		'数列×q，错位相减',
		].join(kbr)
	],


	[['等差⋅等比', 'aq+(a+d)⋅q^2+(a+2d)⋅q^3','+⋯+(a+(n-1)d)⋅q^n'].join(kbr),
		sum('i',1,'n','a_i⋅q^i','','')+kbr+'='+sum('i',1,'n','(a+(i-1)d)⋅q^i','','')+kbr+
		'(1-q)'+sum('i',1,'n','i⋅q^i','','')+kbr+'=aq-a_n⋅q^{n+1}+d'+sum('i',2,'n','q^i','',''),
		[
		Eq([frac('q'+zp('d'+zp(sum('i',1,'n-1','q^i','',''))+'+a_1-a_n⋅q^{n}','[]'),'1-q',''),
		frac('q','1-q','')+'⋅'+zp('d'+zp(frac('q(1-q^{n-1})','1-q',''))+'+a_1-a_n⋅q^{n}','[]'),
		]),
		'数列×q，错位相减'
		].join(kbr)
	],

	[['', 'a+(a+d)⋅q+(a+2d)⋅q^2','+⋯+(a+(n-1)d)⋅q^{n-1}'].join(kbr),
		sum('i',1,'n','a_i⋅q^{i-1}','','')+kbr+'='+sum('i',1,'n','(a+(i-1)d)⋅q^{i-1}','','')+kbr+
		'(1-q)'+sum('i',1,'n','i⋅q^{i-1}','','')+kbr+'=a-a_n⋅q^{n}+d'+sum('i',1,'n-1','q^i','',''),
		[
		Eq([frac('d'+zp(sum('i',1,'n-1','q^i','',''))+'+a_1-a_n⋅q^{n}','1-q',''),
		frac('1','1-q','')+'⋅'+zp('d'+zp(frac('q(1-q^{n-1})','1-q',''))+'+a_1-a_n⋅q^{n}','[]'),
		]),
		'数列×q，错位相减'
		].join(kbr)
	],

	[['', 'a+(a+d)⋅2+(a+2d)⋅2^2','+⋯+(a+(n-1)d)⋅2^{n-1}'].join(kbr),
		sum('i',1,'n','a_i⋅2^{i-1}','','')+kbr+'='+sum('i',1,'n','(a+(i-1)d)⋅2^{i-1}','','')+kbr+
		'(1-2)'+sum('i',1,'n','i⋅2^{i-1}','','')+kbr+'=a-a_n⋅2^{n}+d'+sum('i',1,'n-1','2^i','',''),
		[
		Eq([frac('d'+zp(sum('i',1,'n-1','2^i','',''))+'+a_1-a_n⋅2^{n}','1-q',''),
			'2d+a_{n-1}⋅2^{n}-a',
		]),
		'数列×q，错位相减'
		].join(kbr)
	],

],'wiki').replace(/\n/g,br))+

detail('等差⋅等差 内积求和',Table([ZLR('前n项求和 记法 结果')],[


	['1⋅1+2⋅3+3⋅5+⋯+n(2n-1)',
		Eq([sum('i',1,'n','i(2i-1)','',''),
			'①2'+sum('i',1,'n','i^2','','')+'-'+sum('i',1,'n','i','',''),
		]),

		Eq([
			frac(1,6,'')+'n(n+1)(4n-1)'
		])
		
	],


	['3⋅5+5⋅7+7⋅9+⋯+(2n+1)(2n+3)',
		Eq([sum('i',1,'n','(2i+1)(2i+3)','',''),
			'①4'+sum('i',1,'n','i^2','','')+'+8'+sum('i',1,'n','i','','')+'+3n',
			'②'+frac(1,6,'')+sum('i',1,'n',zp('(2i+1)(2i+3)(2i+5)-(2i-1)(2i+1)(2i+3)'),'',''),

		]),
		Eq([frac(1,6,'')+'((2n+1)(2n+3)(2n+5)-1⋅3⋅5)',
			frac(1,3,'')+'n(4n^2+18n+23)',
		])
	],


	['(a+d)⋅(a+2d)+(a+2d)⋅(a+3d)+⋯+(a+nd)(a+(n+1)d)',
		Eq([sum('i',1,'n','(a+id)(a+id+d)','',''),
			'①d^2'+sum('i',1,'n','i^2','','')+'+(2a+d)d'+sum('i',1,'n','i','','')+'+a(a+d)',
			'②'+frac(1,'3d','')+sum('i',1,'n',zp('(a+id)(a+id+d)(a+id+2d)-(a+id-d)(a+id)(a+id+d)'),'',''),

		]),
		Eq([frac('(a+nd)(a+nd+d)(a+nd+2d)-a(a+d)(a+2d)','3d',''),
			frac(1,3,'')+'n(n^2d^2+3nd(a+d)+3a^2+6ad+2d^2)',
		])
	],


],'wiki').replace(/\n/g,br))+

detail('数列内积和（卷积和）公式',Table([ZLR('前n项求和 记法 结果')],[



	['1⋅2^2+2⋅3^2+3⋅4^2\\\\+⋯+n(n+1)^2',
		sum('i',1,'n','i(i+1)^2','',''),
		Eq([frac(1,12,'')+'n(n+1)(n+2)(3n+5)', frac(1,4,'')+'n(n+1)(n+2)'+zp('n+5\\/3')])
	],

	['1⋅2^2⋅3+2⋅3^2⋅4+3⋅4^2⋅5\\\\+⋯+n(n+1)^2(n+2)',
		sum('i',1,'n','i(i+1)^2(i+2)','',''),
		frac(1,10,'')+'n(n+1)(n+2)(n+3)(2n+3)'
	],


	[['1(n^2-1^2)+','2(n^2-2^2)+','3(n^2-3^2)+','⋯+','(n-1)(n^2-(n-1)^2)'].join(kbr),
		sum('i',1,'n-1','i(n^2-i^2)','',''),
		frac(1,4,'')+'n^2(n^2-1)'
	],


	[['2⋅1⋅2+4⋅2⋅3+8⋅3⋅4','⋯+2^n⋅n(n+1)'].join(kbr),
		sum('i',1,'n','2^i⋅i(i+1)','',''),
		'2^{n+1}(n^2-n+2)-4'
	],




]


	
,'wiki').replace(/\n/g,br))+



detail('阶乘相关求和公式',Table([ZLR('前n项求和 记法 结果')],[



	['1⋅1!+2⋅2!+3⋅3!+⋯+n⋅n!',
		Eq([sum('i',1,'n','i⋅i!','',''),
			sum('i',1,'n',zp('(i+1)!-i!'),'','')
		]),
		Eq([
			'(n+1)!-1'
		])
	],




]


	
,'wiki').replace(/\n/g,br))+

detail('分数和公式',Table([ZLR('前n项求和 记法 结果')],[



	['1\\/{1⋅2}+1\\/{2⋅3}+1\\/{3⋅4}+1\\/{4⋅5}+⋯+1\\/{n(n+1)}',
		Eq([sum('i',1,'n','1\\/{i(i+1)}','',''),
			sum('i',1,'n', zp('1\\/i-1\\/{i+1}'),'',''),
		]),
		Eq(['1-1\\/{n+1}','n\\/{n+1}'])

	],

	['1\\/{k(k+1)}+1\\/{(k+1)(k+2)}+⋯+1\\/{n(n+1)}',
		sum('i','k','n','1\\/{i(i+1)}','','')+kbr+'裂项',
		Eq(['1\\/k-1\\/{n+1}','{n-k+1}\\/{k(n+1)}'])

	],




	['1\\/{1⋅3}+1\\/{2⋅4}+1\\/{3⋅5}+⋯+1\\/{n(n+2)}',
		Eq([sum('i',1,'n','1\\/{i(i+2)}','',''),
			'1\\/2'+sum('i',1,'n', zp('1\\/i-1\\/{i+2}'),'',''),
		]),
		Eq(['1\\/2'+zp('1+1\\/2-1\\/{n+1}-1\\/{n+2}'), 
			'n\\/2'+zp('1\\/{n+1}+1\\/{2(n+2)}'),
			'{n(3n+5)}\\/{4(n+1)(n+2)}'
		])

	],

	['1\\/{k(k+2)}+1\\/{(k+1)(k+3)}+⋯+1\\/{n(n+2)}',
		Eq([sum('i','k','n','1\\/{i(i+2)}','',''),
		'1\\/2'+sum('i','k','n', zp('1\\/i-1\\/{i+2}'),'',''),
	]),
		Eq(['1\\/2'+zp('1\\/k+1\\/{k+1}-1\\/{n+1}-1\\/{n+2}'),
			'{n-k+1}\\/2'+zp('1\\/{k(n+1)}+1\\/{(k+1)(n+2)}')
		])

	],


	['1\\/{1⋅4}+1\\/{2⋅5}+1\\/{3⋅6}+⋯+1\\/{n(n+3)}',
		Eq([sum('i',1,'n','1\\/{i(i+3)}','',''),
			'1\\/3'+sum('i',1,'n', zp('1\\/i-1\\/{i+3}'),'',''),
		]),
		Eq(['1\\/3'+zp('1+1\\/2+1\\/3-1\\/{n+1}-1\\/{n+2}-1\\/{n+3}'),
			'n\\/3'+zp('1\\/{n+1}+1\\/{2(n+2)}+1\\/{3(n+3)}'),
			'{n(11n^2+48n+49)}\\/{18(n+1)(n+2)(n+3)}'
		])

	],

	['1\\/{k(k+3)}+1\\/{(k+1)(k+4)}+⋯+1\\/{n(n+3)}',
		Eq([sum('i','k','n','1\\/{i(i+3)}','',''),
		'1\\/3'+sum('i','k','n', zp('1\\/i-1\\/{i+3}'),'',''),
	]),
		Eq(['1\\/3'+zp('1\\/k+1\\/{k+1}+1\\/{k+2}-1\\/{n+1}-1\\/{n+2}-1\\/{n+3}'),
			'{n-k+1}\\/3'+zp('1\\/{k(n+1)}+1\\/{(k+1)(n+2)}+1\\/{(k+2)(n+3)}')
		])

	],



	['1\\/{1⋅(1+m)}+1\\/{2⋅(2+m)}+⋯+1\\/{n(n+m)}',
		Eq([sum('i',1,'n','1\\/{i(i+m)}','',''),
			'1\\/m'+sum('i',1,'n', zp('1\\/i-1\\/{i+m}'),'',''),
		]),
		Eq(['1\\/m'+zp('1+1\\/2+⋯+1\\/m-1\\/{n+1}-1\\/{n+2}-⋯-1\\/{n+m}'),
			'n\\/m'+zp('1\\/{n+1}+1\\/{2(n+2)}+1\\/{3(n+3)}+⋯+1\\/{m(n+m)}'),
			
		])

	],

	['1\\/{k(k+m)}+1\\/{(k+1)(k+m+1)}+⋯+1\\/{n(n+m)}',
		Eq([sum('i','k','n','1\\/{i(i+3)}','',''),
			'1\\/m'+sum('i','k','n', zp('1\\/i-1\\/{i+m}'),'',''),
		]),
		Eq(['1\\/m'+zp('1\\/k+1\\/{k+1}+⋯+1\\/{k+m}-1\\/{n+1}-1\\/{n+2}-⋯-1\\/{n+m}'),
			'{n-k+1}\\/m'+zp('1\\/{k(n+1)}+1\\/{(k+1)(n+2)}+⋯+1\\/{(k+m-1)(n+m)}'),
		])

	],







	['{3}\\/{1⋅2⋅3}+{5}\\/{2⋅3⋅4}+{7}\\/{3⋅4⋅5}+⋯+{2n+1}\\/{n(n+1)(n+2)}',
		Eq([sum('i',1,'n','{2i+1}\\/{i(i+1)(i+2)}','',''),
			sum('i',1,'n',zp('1\\/{(i+1)(i+2)}+1\\/{i(i+2)}'),'',''),
			sum('i',1,'n',zp('1\\/{i+1}-1\\/{i+2}+1\\/2'+zp('1\\/{i}-1\\/{i+2}')),'',''),
			'1\\/2-1\\/{n+2}+{n(3n+5)}\\/{4(n+1)(n+2)}'

		]),
		Eq(['{n(5n+7)}\\/{4(n+1)(n+2)}'])
	],

	['{2k+1}\\/{k(k+1)(k+2)}+{2n+3}\\/{(k+1)(k+2)(k+3)}+⋯+{2n+1}\\/{n(n+1)(n+2)}',
		Eq([sum('i','k','n','{2i+1}\\/{i(i+1)(i+2)}','',''),
			sum('i','k','n',zp('1\\/{(i+1)(i+2)}+1\\/{i(i+2)}'),'',''),
			sum('i','k','n',zp('1\\/{i+1}-1\\/{i+2}+1\\/2'+zp('1\\/{i}-1\\/{i+2}')),'',''),
			'1\\/{k+1}-1\\/{n+2}+1\\/2'+zp('1\\/k+1\\/{k+1}-1\\/{n+1}-1\\/{n+2}'),
			'1\\/2'+zp('1\\/k+3\\/{k+1}-1\\/{n+1}-3\\/{n+2}')

		]),
		Eq(['{n-k+1}\\/2'+zp('1\\/{k(n+1)}+3\\/{(k+1)(n+2)}')])
	],



	['{1}\\/{1⋅2⋅3}+{1}\\/{2⋅3⋅4}+{1}\\/{3⋅4⋅5}+⋯+{1}\\/{n(n+1)(n+2)}',
		Eq([sum('i',1,'n','{1}\\/{i(i+1)(i+2)}','',''),
			'1\\/2'+sum('i',1,'n',zp('1\\/{i(i+1)}-1\\/{(i+1)(i+2)}'),'',''),
			'①1\\/2'+zp('1\\/2-1\\/{(n+1)(n+2)}'),
			'②1\\/2'+zp(sum('i',1,'n','1\\/{i(i+1)}','','')+'-'+sum('i',1,'n','1\\/{(i+1)(i+2)}','','')),
			
			'1\\/2'+zp('1-1\\/{n+1}-'+zp('1\\/2-1\\/{n+2}')),
		]),
		Eq(['1\\/2'+zp('1\\/2-1\\/{(n+1)(n+2)}'),
			'{n(n+3)}\\/{4(n+1)(n+2)}',
		])
	],




	['{1}\\/{1⋅2⋅3⋅4}+{1}\\/{2⋅3⋅4⋅5}+{1}\\/{3⋅4⋅5⋅6}+⋯+{1}\\/{n(n+1)(n+2)(n+3)}',
		Eq([sum('i',1,'n','{1}\\/{i(i+1)(i+2)(i+3)}','',''),
			'1\\/2'+sum('i',1,'n',zp('1\\/{i(i+3)}-1\\/{(i+1)(i+2)}'),'',''),
			'1\\/2'+sum('i',1,'n',zp('1\\/3'+zp('1\\/i-1\\/{i+3}')+'+1\\/{i+2}-1\\/{i+1}'),'',''),
			
			'1\\/2'+zp('1\\/{n+2}-1\\/2+'+sum('i',1,'n',zp('1\\/3'+zp('1\\/i-1\\/{i+3}')+'+1\\/{i+2}-1\\/{i+1}'),'','')),

		]),
		Eq(['{n(5n+7)}\\/{4(n+1)(n+2)}'])
	],


]


	
,'wiki').replace(/\n/g,br))+




detail('分数和公式（涉及平方）',Table([ZLR('前n项求和 记法 结果')],[

	['{1^2+2^2}\\/{1⋅2}+{2^2+3^2}\\/{2⋅3}+{3^2+4^2}\\/{3⋅4}+⋯+{n^2+(n+1)^2}\\/{n(n+1)}',
		Eq([sum('i',1,'n','{i^2+(i+1)^2}\\/{i(i+1)}','',''),
			sum('i',1,'n',zp('i\\/{i+1}+{i+1}\\/i'),'',''),
			sum('i',1,'n',zp('2+1\\/i-1\\/{i+1}'),'',''),

		]),
		Eq(['2n+1-1\\/{n+1}','2n+n\\/{n+1}','n'+zp('2+1\\/{n+1}')])
	],

	['{2^2}\\/{1⋅3}+{4^2}\\/{3⋅5}+{6^2}\\/{5⋅7}+⋯+{(2n)^2}\\/{(2n-1)(2n+1)}',
		Eq([sum('i',1,'n','{(2i)^2}\\/{(2i-1)(2i+1)}','',''),
			sum('i',1,'n',zp('1+1\\/{(2i-1)(2i+1)}'),'',''),
			sum('i',1,'n',zp('1+1\\/2'+zp('1\\/{2i-1}-1\\/{2i+1}')),'',''),

		]),
		Eq(['n+1\\/2'+zp('1-1\\/{2n+1}'), '{2n(n+1)}\\/{2n+1}'])
	],






	['{3}\\/{(1⋅2)^2}+{5}\\/{(2⋅3)^2}+{7}\\/{(3⋅4)^2}+⋯+{2n+1}\\/{(n(n+1))^2}',
		Eq([sum('i',1,'n','{2i+1}\\/{(i(i+1))^2}','',''),
			sum('i',1,'n','{i+(i+1)}\\/{(i(i+1))^2}','',''),
			sum('i',1,'n',zp('1\\/{i^2}-1\\/{(i+1)^2}'),'',''),

		]),
		Eq(['1-1\\/{(n+1)^2}','{n(n+2)}\\/{(n+1)^2}'])
	],





]


	
,'wiki').replace(/\n/g,br))+



detail('交错分数和公式',Table([ZLR('前n项求和 记法 结果')],[



	['{3}\\/{1⋅2}-{5}\\/{2⋅3}+{7}\\/{3⋅4}-⋯-{4n+1}\\/{2n(2n+1)}',
		Eq([sum('i',1,'2n','\\frac{(2i+1)(-1)^{i+1}}{i(i+1)}','',''),
			'①'+sum('i',1,'n',zp('{4i-1}\\/{(2i-1)(2i)}-{4i+1}\\/{(2i)(2i+1)}'),'',''),
			sum('i',1,'n','{2}\\/{(2i-1)(2i+1)}','',''),
			sum('i',1,'n',zp('1\\/{2i-1}-1\\/{2i+1}'),'',''),

			'②'+sum('i',1,'2n','(-1)^{i+1}'+zp('1/i+1\\/{(i+1)}'),'',''),
			sum('i',1,'n',zp('1\\/{2i-1}+1\\/{2i}-1\\/{2i}-1\\/{(2i+1)}'),'',''),
			sum('i',1,'n',zp('1\\/{2i-1}-1\\/{2i+1}'),'',''),


		]),
		Eq(['1-1\\/{2n+1}','{2n}\\/{2n+1}'])
	],

	['{5}\\/{2⋅3}-{7}\\/{3⋅4}+{9}\\/{4⋅5}-⋯-{4n+3}\\/{(2n+1)(2n+2)}',
		Eq([sum('i',1,'2n','\\frac{(2i+3)(-1)^{i+1}}{(i+1)(i+2)}','',''),
			sum('i',1,'n', zp('{4i+1}\\/{(2i)(2i+1)}-{4i+3}\\/{(2i+1)(2i+2)}'),'',''),
			sum('i',1,'n','1\\/{2i(i+1)}','',''),
			'{3}\\/{1⋅2}-{4n+3}\\/{(2n+1)(2n+2)}-'+sum('i',1,'2n','\\frac{(2i+1)(-1)^{i+1}}{i(i+1)}','','')
		]),
		Eq(['1\\/2'+zp('1-1\\/{n+1}'),'{n}\\/{2(n+1)}',
		])
	],

]


	
,'wiki').replace(/\n/g,br))+





detail('膨胀分数和公式',Table([ZLR('前n项求和 记法 结果')],[



	['1+1\\/{1+2}+1\\/{1+2+3}+⋯+1\\/{1+2+⋯+n}',
		Eq([sum('i',1,'n','1\\/{1+2+⋯+i}','',''),
			sum('i',1,'n','2\\/{i(i+1)}','',''),
			sum('i',1,'n',2+zp('1\\/i-1\\/{i+1}'),'',''),

		]),
		Eq([2+zp('1-1\\/{n+1}'),'{2n}\\/{n+1}'])
	],


	['1+{1^3+2^3}\\/{1+2}+{1^3+2^3+3^3}\\/{1+2+3}+⋯+{1^3+2^3+⋯+n^3}\\/{1+2+⋯+n}',
		Eq([sum('i',1,'n','{1^3+2^3+⋯+i^3}\\/{1+2+⋯+i}','',''),
			sum('i',1,'n','\\frac{'+sum('j',1,'i','j^3','','')+'}{'+sum('j',1,'i','j','','')+'}','',''),
			sum('i',1,'n',sum('j',1,'i','j','',''),'',''),
			sum('i',1,'n','\\frac{i(i+1)}{2}','',''),
			'1\\/2'+sum('i',1,'n','i(i+1)','',''),
		]),
		Eq(['{n(n+1)(n+2)}\\/6'])
	],

	
	['1\\/{1⋅2}+2\\/{1⋅2⋅3}+3\\/{1⋅2⋅3⋅4}+⋯+n\\/{1⋅2⋯n(n+1)}',
		Eq([sum('i',1,'n','i\\/{(i+1)!}','',''),
			sum('i',1,'n',zp('1\\/{i!}-1\\/{(i+1)!}'),'',''),

		]),
		Eq(['1-1\\/{(n+1)!}'])
	],


	['n\\/{n-2}+{n(n-1)}\\/{(n-2)(n-3)}+{n(n-1)(n-2)}\\/{(n-2)(n-3)(n-4)}'+kbr+'+⋯+{n(n-1)(n-2)⋯4⋅3}\\/{(n-2)(n-3)(n-4)⋯2⋅1}',
		Eq(['①'+sum('i',1,'n-2','{n(n-1)⋯(n-i+1)}\\/{(n-2)(n-3)⋯(n-i-1)}','',''),
			sum('i',1,'n-2','{n!/(n-i)!}\\/{(n-2)!/(n-i-2)!}','',''),
			sum('i',1,'n-2','{n!}\\/{(n-2)!} {(n-i-2)!}\\/{(n-i)!}','',''), 

			sum('i',1,'n-2','{n(n-1)}\\/{(n-i)(n-i-1)}','',''),
			'n(n-1)'+sum('i',1,'n-2', zp('1\\/{n-i-1}-1\\/{n-i}'),'',''),

			'②'+sum('i',1,'n-2','{n(n-1)⋯(i+2)}\\/{(n-2)(n-3)⋯i}','',''),
			sum('i',1,'n-2','{n!/(i+1)!}\\/{(n-2)!/(i-1)!}','',''),
			sum('i',1,'n-2','{n!}\\/{(n-2)!} {(i-1)!}\\/{(i+1)!}','',''), 
			sum('i',1,'n-2','{n(n-1)}\\/{i(i+1)}','',''),
			'n(n-1)'+sum('i',1,'n-2',zp('1\\/{i}-1\\/{i+1}'),'',''),

		]),
		Eq(['n(n-2)'])
	],

	['n\\/{n-k}+{n(n-1)}\\/{(n-k)(n-k-1)}+{n(n-1)(n-2)}\\/{(n-k)(n-k-1)(n-k-2)}'+kbr+'+⋯+{n(n-1)(n-2)⋯(k+2)(k+1)}\\/{(n-k)(n-k-1)(n-k-2)⋯2⋅1}',
		Eq(['①'+sum('i',1,'n-k','{n(n-1)⋯(n-i+1)}\\/{(n-k)(n-k-1)⋯(n-k-i+1)}','',''),
			sum('i',1,'n-k','{n!/(n-i)!}\\/{(n-k)!/(n-k-i)!}','',''),
			sum('i',1,'n-k','{n!}\\/{(n-k)!} {(n-k-i)!}\\/{(n-i)!}','',''), 

			sum('i',1,'n-k','{n(n-1)⋯(n-k+1)}\\/{(n-i)(n-i-1)⋯(n-i-k+1)}','',''),
			'n(n-1)⋯(n-k+1)'+sum('i',1,'n-k','1\\/{(n-i)(n-i-1)⋯(n-i-k+1)}','',''),

			'②'+sum('i',1,'n-k','{n(n-1)⋯(i+k)}\\/{(n-k)(n-k-1)⋯i}','',''),
			sum('i',1,'n-k','{n!/(i+k-1)!}\\/{(n-k)!/(i-1)!}','',''),
			sum('i',1,'n-k','{n!}\\/{(n-k)!} {(i-1)!}\\/{(i+k-1)!}','',''), 
			sum('i',1,'n-k','{n(n-1)⋯(n-k+1)}\\/{(i+k-1)(i+k-2)⋯i}','',''),
			'n(n-1)⋯(n-k+1)'+sum('i',1,'n-k','1\\/{(i+k-1)(i+k-2)⋯i}','',''),

		]),
		Eq([''])
	],
]


	
,'wiki').replace(/\n/g,br))+




detail(i18('Reference'),ol([

	enwiki("Catalan's_conjecture"),
	enwiki('Diophantine_equation','2020-8-4'),

	enwiki('Umbral_calculus','2019-11-22'),
	enwiki("Sums_of_powers",'2020-8-11'),
	enwiki("Bernoulli_number",'2020-10-15'),
	enwiki("Bernoulli_polynomials",'2020-10-16'),
	enwiki("Faulhaber's_formula",'2020-10-15'),
	enwiki("Squared_triangular_number",'2020-10-15'),
	enwiki("Q-analog",'2020-10-15'),

	inhref('wiki.html?q=Formula/Equation/Diophantus'),
	inhref('wiki.html?q=Concept/Number/Sequence/Rational'),
	inhref('explore.html?q=Problem/Problem List'),
	'《数学手册》，高等教育出版社，1979.5 第1版',
	'《实用数学手册》，科学出版社，2006.1 第2版',
	'《数学指南——实用数学手册》，科学出版社，2012.1 第1版',
]))



